Curvature-slope correlation of nuclear symmetry energy and its imprints on the crust-core transition, radius and tidal deformability of canonical neutron stars
aa r X i v : . [ nu c l - t h ] O c t Curvature-slope correlation of nuclear symmetry energy and its imprints on thecrust-core transition, radius and tidal deformability of canonical neutron stars
Bao-An Li ∗ and Macon Magno † Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429, USA (Dated: October 6, 2020)
Background:
The nuclear symmetry energy E sym ( ρ ) encodes information about the energy necessary to make nuclear systemsmore neutron-rich. It is currently poorly known especially at supra-saturation densities but has broad impacts on proper-ties of neutron stars, nuclear structures and reactions. While its slope parameter L at the saturation density ρ of nuclearmatter has been relatively well constrained by recent astrophysical observations and terrestrial nuclear experiments, itscurvature K sym characterizing the E sym ( ρ ) around 2 ρ remains largely unconstrained. Over 520 calculations for E sym ( ρ )using various nuclear theories and interactions in the literature have predicted several significantly different K sym − L correlations. Purpose:
If a unique K sym − L correlation of E sym ( ρ ) can be firmly established, it will enable us to progressively constrain thehigh-density behavior of E sym ( ρ ) using the available and better constrained slope parameter L. We investigate if and byhow much the different K sym − L correlations may affect neutron star observables. We also examine if LIGO/VIRGO’sobservation of tidal deformability using gravitational waves from GW170817 and NICER’s recent extraction of neutronstar radius using high-precision X-rays can distinguish the different K sym − L correlations predicted. Method:
A meta-model of nuclear Equation of States (EOSs) with three representative K sym − L correlation functions is usedto generate multiple EOSs for neutron stars. We then examine effects of the K sym − L correlation on the crust-coretransition density and pressure as well as the radius and tidal deformation of canonical neutron stars. Results:
We found that the K sym − L correlation affects significantly both the crust-core transition density and pressure. Italso has strong imprints on the radius and tidal deformability of canonical neutron stars especially at small L values.The available data from LIGO/VIRGO and NICER set some useful limits for the slope L but can not distinguish thethree representative K sym − L correlations considered. Conclusions:
The K sym − L correlation is important for understanding properties of neutron stars. More precise and preferablyindependent measurements of the radius and tidal deformability from multiple observables of neutron stars have the strongpotential to help pin down the curvature-slope correlation, thus the high-density behavior of nuclear symmetry energy. I. INTRODUCTION
The crust-core transition density and pressure in neu-tron stars (NSs) play significant roles in modeling NSobservational properties [1, 2]. In particular, they af-fect the fractional moment of inertia of a NS’s crustclosely related to the pulsar glitch phenomenon as wellas the radius and quadrupole deformation of both iso-lated NSs and those involved in NS binary mergers. Re-gardless of the approaches used, determining the NScrust-core transition point involves both the first-orderand second-order derivatives of the nucleon average en-ergy in isospin-asymmetric nuclear matter with respectto the density of neutrons and protons. It thus has beenknown since the earlier 70’s that the core-crust transi-tion density and pressure are very sensitive to the finedetails of the isospin and dense dependences of the nu-clear Equation of State (EOS) [1–3]. Indeed, extensivestudies using various nuclear many-body theories andinteractions have examined effects of various terms inthe EOS and demonstrated the significant model depen-dence of the NS crust-core transition density and pres-sure, see, e.g., Ref. [4] for a recent review. In particular, ∗ [email protected] † [email protected] both the slope L = 3 ρ [ ∂E sym ( ρ ) /∂ρ ] | ρ = ρ and curvature K sym = 9 ρ [ ∂ E sym ( ρ ) /∂ρ ] | ρ = ρ of nuclear symmetryenergy E sym ( ρ ) at the saturation density ρ of nuclearmatter were found to play very important roles in de-termining the crust-core transition density and pressure.They both also affect significantly the radii and tidal de-formations of canonical neutron stars [4]. In fact, it iswell known that NS radii are mostly determined by thenuclear pressure around 2 ρ where the K sym has the mostimportant influence on the E sym ( ρ ) and the correspond-ing pressure there [5–8]. In turn, imprints of K sym onobservational properties of NSs may help us further con-strain the poorly known high-density behavior of nuclearsymmetry energy.Unfortunately, it has been very challenging to con-strain the density dependence of nuclear symmetry en-ergy E sym ( ρ ) especially at supra-saturation densities [9].While some significant progresses have been made in ex-perimentally constraining the slope parameter L char-acterizing mostly the E sym ( ρ ) around ρ , the curvature K sym dominating the behavior of E sym ( ρ ) around 2 ρ ismuch less constrained presently. More quantitatively, asurvey in 2013 of 28 previous analyses of terrestrial nu-clear laboratory experiments and astrophysical observa-tions found a fiducial value of L = 59 ±
16 MeV [10]. Itchanged to L = 58 . ± . K sym . On the otherhand, extensive surveys of over 520 theoretical predic-tions available in the literature up to 2014 [12, 13] indi-cated that K sym is in the range of − ≤ K sym ≤ L = 66 +12 − and K sym = − +80 − at 68% confidence level [19, 20]. The extracted L isconsistent with its fiducial value within the error bars.These analyses of the new astrophysical data clearly pro-vided some useful constraints on the model predictionsfor K sym , but its uncertainty range remains quite large.Predictions of the nuclear EOS based on many-bodytheories subject to various constraints, such as empir-ical properties of nuclear matter at ρ , some proper-ties of finite nuclei and/or terrestrial nuclear reactions,naturally introduce correlations among some features ofthe EOSs predicted. Indeed, strong correlations be-tween L and K sym have been found in predictions byvarious nuclear energy density functionals and/or micro-scopic many-body theories, see, e.g., Refs. [12–15, 21–26]. As discussed above, some reasonably tight experi-mental/observational constraints on the slope L alreadyexist. Moreover, coming nuclear experiments as well asastrophysical observations will help narrow down furtherits uncertainties. Thus, if one can establish firmly the K sym − L correlation, it can then help progressively con-strain the K sym using the available constraints on L.In this work, using the EOS meta-modeling approachof Refs. [27–30] we study effects of the K sym − L corre-lation on the crust-core transition density and pressureas well as its imprints on the radius and tidal deforma-bility of canonical neutron stars of mass 1.4 M ⊙ . Whennecessary and possible, we also discuss if constraints onthe NS tidal deformability from LIGO/VIRGO’s observa-tion of GW170817 [16] and constraints on the NS radiusfrom NICER’s recent observation of PSR J0030+0451[17, 18] can help distinguish different K sym − L correla-tions and/or how they may help constrain the L param-eter.The rest of the paper is organized as follows. In thenext section, we first outline the EOS meta-modelingmethod and three representative K sym − L correlationfunctions available in the literature. We then examine ef-fects of the K sym − L correlation on the crust-core transi-tion density and pressure by comparing results using thethree different K sym − L correlation functions in a largeEOS parameter space allowed by all existing constraints.We then investigate imprints of the K sym − L correlationon the radius and tidal deformability of neutron stars. Asummary and conclusions are given at the end. II. THEORETICAL APPROACH
Here we outline the major components of our ap-proach. We focus on the new features but skip mostof the details that one can easily find in the literature.For completeness and ease of the following discussions,when necessary we also recall briefly some of the wellestablished equations and methods we adopted here.
A. Parameterizing the EOS of neutron-richnucleonic matter
For neutron-rich nucleonic matter of neutron density ρ n and proton density ρ p , it has an isospin asymmetry δ = ( ρ n − ρ p ) /ρ and density ρ = ρ n + ρ p . Its EOS canbe written as [31] E ( ρ, δ ) = E ( ρ ) + E sym ( ρ ) δ + O ( δ ) (1)in terms of the energy per nucleon E ( ρ ) ≡ E ( ρ, δ = 0)in symmetric nuclear matter (SNM) and the symmetryenergy E sym ( ρ ). For a given EOS E ( ρ, δ ) from a nuclearmany-body theory, it is customary to Taylor expand boththe E ( ρ ) and E sym ( ρ ) as functions of ( ρ − ρ )(3 ρ ) withcoefficients given by their density derivatives at ρ . Thisapproach is particularly useful in the traditional forward-modeling of various physics problems. Unfortunately, thecoefficients predicted so far are still very model depen-dent and often show characteristically different correla-tions.In a different method that is almost opposite to the tra-ditional approach mentioned above, independent of thenuclear many-body theories and interactions used andwithout knowing a priori the EOS, one can simply pa-rameterize the SNM EOS E ( ρ ) and the symmetry en-ergy E sym ( ρ ) as functions of ( ρ − ρ ) / (3 ρ ) in the sameform as if they are Taylor expansions of some knownEOSs. By randomly generating the relevant parametersfor the parameterized E ( ρ ) and E sym ( ρ ), one can mimicall available EOSs in the literature. By purposely pa-rameterizing the E ( ρ ) and E sym ( ρ ) as if they are Taylorexpansions, one can limit their parameter ranges to thoseof the Taylor coefficients predicted by all available nu-clear many-body theories. Such kinds of meta-modelingof nuclear EOSs [32, 33] or EOS generators [27–30] havebeen found particularly useful in solving the NS inverse-structure problems. They have been used successfully inboth the direct inversion of NS observables in the three-dimensional high-density EOS parameter space [27–30]and the Bayesian inferences of EOS parameters from NSobservables [19, 20, 34, 35] or heavy-ion reaction data[36]. In general, parameterized functions are necessaryin all machine learning processes. It is advantageousto select functions such that one can make better useof available data/knowledge to set the prior ranges andprobability distributions of the parameters. Our param-eterizations of the E ( ρ ) and E sym ( ρ ) are based on thisconsideration.We parameterize the E ( ρ ) and E sym ( ρ ) up to the thirdpower of ( ρ − ρ )(3 ρ ) according to E ( ρ ) = E ( ρ ) + K ρ − ρ ρ ) + J ρ − ρ ρ ) , (2) E sym ( ρ ) = E sym ( ρ ) + L ( ρ − ρ ρ ) + K sym ρ − ρ ρ ) + J sym ρ − ρ ρ ) . (3)Values of E ( ρ ) and E sym ( ρ ) have no effect on thecrust-core transition and have little effects on NS globalproperties. We thus fix them at their known empiricalvalues of E ( ρ ) = − . ± . E sym ( ρ ) =31 . ± . K , J , L, K sym and J sym are varied in ranges or fixed at specificvalues consistent with our currently knowledge from nu-clear theories and experiments. For instance, the mostprobable incompressibility of symmetric nuclear matteris relatively well constrained to K = 240 ±
20 MeV [37–39]. We thus use three separate values of 220, 240 and 260MeV for the K . The J was predicted to be in the rangeof − ≤ J ≤
400 MeV using various nuclear theoriesand forces [14, 15]. Its most probable value was found tobe J = − +55 − [20] at 68% confidence level from veryrecent Bayesian analyses of NS properties, while a valueof J = − +20 − MeV [36] was inferred from a recentBayesian analysis of the collective flow and kaon produc-tion in relativistic heavy-ion collisions. Obviously, theseBayesian analyses have clearly narrowed down the pre-dicted range of J , but still have relatively large errors.Fortunately, as it has been shown very recently in Ref.[35] in a Bayesian analysis using the EOS meta-modelingapproach of Refs. [32, 33], the crust-core transition den-sity and pressure are insensitive to J even when it isvaried in an extremely large range between − J = 0. While in constructing the coreEOS for studying properties of NSs, we will fix it at avalue sufficient to support NSs with a maximum mass ofabout 2 M ⊙ , and be consistent with the results of thetwo Bayesian analyses mentioned above.Consistent with earlier findings, it was also shown inRef. [35] that the crust-core transition density and pres-sure are very sensitive to the symmetry energy param-eters by varying them independently within their largeprior ranges of 10 ≤ L ≤
80 MeV, − ≤ K sym ≤ − ≤ J sym ≤ L and curvature K sym while the sensitivity to the J sym parameter is weaker but appreciable. These useful priorknowledge and finings are considered when we vary theEOS parameters in our own study here. B. Slope-curvature correlation of nuclearsymmetry energy
Among the K sym − L correlation functions found in theliterature, the following one by Mondal et al. [25] K sym = ( − . ± . E sym ( ρ ) − L )+66 . ± .
14 MeV(4)is based on probably the most number of theoretical pre-dictions including 240 Skyrme Hartree-Fock (SHF) [12]and 263 Relativistic Mean-Field (RMF) calculations [13]compiled by Dutra et al. . Using the same inputs but re-stricting to predictions giving 0 . < ρ < .
17 fm − , − < E ( ρ ) < −
15 MeV, 25 < E sym ( ρ ) <
36 MeV,and 180 < K <
275 MeV, Tews et al. [14] deduced thefollowing correlation at 68% confidence level K sym = 3 . L − . ± .
26 MeV . (5)Since the above two correlations stem from the same setsof model predictions albeit some additional selection cri-teria were used by Tews et al. , as shown in Fig. 1, theylargely overlap for L >
60 MeV but show significant dif-ferences at lower L values.
40 45 50 55 60 65 70 75 80L [MeV]-250-200-150-100-50050 K s y m [ M e V ] Mondal et al. (2017)Holt et al. (2018)Tews et al. (2017)
FIG. 1: (Color online) The K sym − L correlation from Tews etal. [14] (red), Mondal et al. [25] (black) and Holt et al. [26](green), respectively. The solid lines are the means and thedashed lines are the upper and lower limits of each correlation. More recently, within the Fermi liquid theory with pa-rameters benchmarked by chiral effective field theory pre-dictions at sub-saturation densities, Holt and Lim [26]derived the relations L = 6 . E sym ( ρ ) − . ± . K sym = 18 . E sym ( ρ ) − . ± .
62 MeV,leading to the K sym − L correlation of K sym = 2 . L − . ± .
69 MeV . (6)This correlation is shown as the green line in Fig. 1.In the range of 40 < L <
80 MeV consistent with itsfiducial range from earlier surveys [10, 11], the range of K sym especially at lower L values in the Holt correlationis significantly higher than those in the other two cases.As we shall show, this has significant effects on proper-ties of neutron stars. Holt et al. found that the largestsource of uncertainty in their K sym − L correlation is theassumed fiducial value of K between 220 and 260 MeV.We will thus also examine the role of K in comparisonwith that of the K sym − L correlation on the crust-coretransition and radius of canonical neutron stars. We notethat in this study we only used the means of the K sym − L correlations shown in Fig. 1 without considering the un-certainty of each individual correlation. C. Thermodynamical method for finding thecrust-core transition density in neutron stars
Since the pioneering work of Baym et al. in 1971 [1, 2],the crust-core transition density and pressure in NSs havebeen studied extensively using several approaches start-ing from either the crust or core side. The most widelyused one is by examining whether small density fluctua-tions will grow in the uniform core. This is often done byusing the dynamical method considering the surface andCoulomb effects of clusters or its long-wavelength limit,i.e., the thermodynamical method, see, e.g., Refs. [43–70], or the RPA [71–73]. The crust-core transition hasalso been studied by comparing the free energy of clus-tered matter with that of the uniform matter either us-ing various mass models within the Compressible LiquidDrop Mode [1, 2, 43, 44, 49, 74–76] or the 3D Hartree-Fock theory [77, 78] for nuclei on the Coulomb latticeusing the Wigner-Seitz approximation.Perhaps, the simplest approach is the thermodynam-ical method which we use here. In this approach[7, 45, 46], the crust-core transition density is found byexamining when the following effective incompressibilityof the uniform NS core at β -equilibrium becomes negative(the corresponding speed of sound becomes imaginary),indicating the start of cluster formation (or the onset ofspinodal decomposition) K µ = ρ d E dρ + 2 ρ dE dρ (7)+ δ " ρ d E sym dρ + 2 ρ dE sym dρ − E − ( ρ ) (cid:18) ρ dE sym dρ (cid:19) . In terms of several EOS parameters, the crust-core tran-sition density is determined by setting K µ = 19 ( ρρ ) K + 2 ρ dE dρ (8)+ δ (cid:20)
19 ( ρρ ) K sym + 23 ρρ L − E − ( ρ )( 13 ρρ L ) (cid:21) = 0 . The last two terms in the bracket (isospin-dependentpart) approximately cancel out, thus leaving the K sym dominates [27]. Nevertheless, the K and L also play significant roles. Thus, different K sym − L correlationsare expected to affect the crust-core transition density.For earlier discussions on this issue, we refer the readerto Ref. [4, 23, 76] and the references therein.It is interesting to note that Eq. (8) may have a sec-ond solution at a supra-saturation density besides theone at a sub-saturation density indicating the crust-coretransition [46]. This happens only in cases where thesymmetry energy is super-soft (flat or decreasing withincreasing density) at high densities when the L is verysmall but the K sym has a big negative value. In thesecases, if the incompressibility K of SNM is not highenough, then the negative contribution of the symmetryenergy to the K µ may cause the latter to decrease withincreasing density. At some critical density, it will thenreach zero again, indicating the onset of another dynami-cal instability. While by linking the liquid core EOS witha crust EOS at the crust-core transition density one hasconstructed a stable EOS up to the onset of the secondinstability, the physical meaning of the latter is currentlynot clear. Interestingly, it was speculated in Ref. [46]that the second instability may indicate the start of an-other new phase, e.g., solidification, in the inner core ofneutron stars. Without knowing how to model this essen-tially pure neutron matter core (as a result of the super-soft symmetry energy) at very high densities (above thegenerally expected hadron-quark transition density), wesimply stop generating the EOS when the second instabil-ity happens (by default in the code by enforcing the dy-namical stability condition in meta-modeling the EOS).This can happen before the causal limit or the dM/dR=0point on the mass (M)-radius (R) curve is reached.We found that the second instability happens mostlywith the Mondal and Tews correlations giving large neg-ative values of K sym when L is around 40-50 MeV and K is around 220-240 MeV. It does not happen with theHolt correlation as it gives significantly higher K sym val-ues. These can be easily understood from the competi-tion of the different terms in Eq. (8). Thus, the threedifferent K sym -L correlations with small L values affectsignificantly the maximum masses they predict. How-ever, they have little effects on properties of canonicalneutron stars as the second instability happens normallyat very high densities reachable only in the core of mas-sive neutron stars. For example, for the Mondal cor-relation with L=40 MeV, K sym =-205 MeV, J sym =296.8MeV, K = 220 MeV and J = −
180 MeV, the secondinstability happens at 6 . ρ . As a result, the maximummass this EOS parameter set can support is about 1.9M ⊙ while that with the Holt correlation is about 2.1 M ⊙ as we shall discuss in more detail in Section III C.In our opinion, the possible appearance of the secondinstability when the symmetry energy is super-soft in theinner core of neutron stars is not a deficiency of the EOSmeta-model we used. As pointed out in Ref. [46], theremight be interesting new physics associated with the sec-ond instability. Without restrictions of the underlyingenergy density functionals in various nuclear many-bodytheories, the EOS meta-model can freely explore the en-tire EOS parameter space allowed by general physicsprinciples. It can thus facilitate the study of previouslyunexplored areas of the EOS parameter space and thecorresponding phases of neutron star matter. Since thesecond dynamical instability may happen mostly at den-sities above the normally expected hadron-quark tran-sition and we do not have a model for the EOS of thepossible new phase above the second instability, we post-pone the study on the possibly new physics associatedwith the latter to a future work. In this work, within the npeµ model enforcing the dynamical stability throughoutneutron stars we focus on properties of canonical neutronstars that are not affected by the possible second insta-bility. D. Constructing the EOS for neutron stars
Within the npeµ model assuming NSs are made of neu-trons, protons, electrons and muons at β -equilibrium un-der charge neutrality and dynamical stability conditions,the pressure P ( ρ, δ ) = ρ dǫ ( ρ, δ ) /ρdρ (9)is obtained from the energy density ǫ ( ρ, δ ) = ǫ n ( ρ, δ ) + ǫ l ( ρ, δ ) with ǫ n ( ρ, δ ) and ǫ l ( ρ, δ ) being the energy den-sities of nucleons and leptons, respectively. While the ǫ l ( ρ, δ ) is calculated using the noninteracting Fermi gasmodel [83], the ǫ n ( ρ, δ ) is from ǫ n ( ρ, δ ) = ρ [ E ( ρ, δ ) + M N ] (10)where M N is the average nucleon mass.Below the crust-core transition density, we use the NVEOS [79] for the inner crust and the BPS EOS [1] for theouter crust. For the purposes of this work, this choice issufficient. However, we notice that more modern descrip-tions for both the inner crust including a possible pastaphase, see, e.g. [76], and the outer crust built from thesame interaction as the core EOS in a uniform approach,see, e.g. [80], are available in the literature.Having discussed earlier how to find the crust-coretransition point, we now discuss briefly how the core EOSis determined, namely selecting the values or ranges ofthe high-density EOS parameters especially the J and J sym . The maximum mass 2.14 M ⊙ of NSs observed sofar [81] requires J to be higher than about −
200 MeV[30] slightly depending on the symmetry energy param-eters L , K sym and J sym used [30]. For the purposes ofthis work focusing on effects of K sym − L correlation onproperties of canonical NSs, it is sufficient to simply usea constant J that is large enough to support NSs asmassive as about 2.0 M ⊙ in the whole EOS parameterspace considered. Here we present results all obtainedwith J = −
180 MeV. Using different values, e.g., -215MeV, also satisfying the above conditions and being con-sistent with the results of Bayesian analyses of both NS properties and heavy-ion reactions mentioned earlier, ourresults remain qualitatively the same.To our best knowledge, presently there is no clear con-straint on the value of J sym from neither astrophysicalobservations not terrestrial experiments. Interestingly,however, the study of Mondal et al. predicted its mostprobable value at J sym = 296 . ± . K sym − L correlations, herewe adopt the value of J sym = 296 . J sym . Wedid systematically vary the values of both J and J sym under the conditions that all EOSs have to be casual,dynamically stable and stiff enough to support NSs asmassive as about 2.0 M ⊙ , our qualitative conclusions arethe same while there are some slight differences quanti-tatively. Since the crust-core transition density and pres-sure have some appreciable dependences on J sym , to beconsistent, we use the same J sym in finding the crust-coretransition point and constructing the core EOS unlessotherwise specified.The resulting EOS for the whole NS in the form of P ( ǫ )is then used as the input in solving the standard Tolman-Oppenheimer-Volkov (TOV) NS structure equations [82,83] dPdr = − G ( m ( r ) + 4 πr P/c )( ǫ + P/c ) r ( r − Gm ( r ) /c ) , (11) dm ( r ) dr = 4 πǫr . (12)The NS mass M is obtained from integrating the massprofile m ( r ) and the radius R is found when the pressurebecomes zero on the surface starting from the centralpressure P c where m (0) = 0. The codes used in thiswork are developed from modifying those used in Refs.[27, 84]. III. RESULTS AND DISCUSSIONSA. Effects of the K sym − L correlation on the NScrust-core transition density Shown in Fig. 2 are the crust-core transition den-sity as a function of L with the indicated three differ-ent K sym − L correlations, K = 220 ,
240 and 260 MeV, J sym (crust) = −
200 (left) and +296 . K sym instead. Overall, the crust-coretransition density is around ρ / K sym − L correlation has the strongest effecton the crust-core transition density.
40 45 50 55 60 65 70 75 80L [MeV]0.060.070.080.090.10.11 ρ t [f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017) K = 220 MeVK = 240 MeVK = 260 MeVJ sym (crust) = -200 MeV 40 45 50 55 60 65 70 75 80L [MeV]0.060.070.080.090.10.11 ρ t [f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017) K = 220 MeVK = 240 MeVK = 260 MeVJ sym (crust) = 296.8 MeV FIG. 2: (Color online) The crust-core transition density as a function of L with the indicated three different K sym − L correlationswith K = 220 ,
240 and 260 MeV and J sym (crust) = −
200 (left) and +296 . -225 -200 -175 -150 -125 -100 -75 -50 -25 0 25K sym [MeV]0.060.070.080.090.10.11 ρ t [f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017) K = 220 MeVK = 240 MeVK = 260 MeVJ sym (crust) = -200 MeV -225 -200 -175 -150 -125 -100 -75 -50 -25 0 25K sym [MeV]0.060.070.080.090.10.11 ρ t [f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017)K = 220 MeVK = 240 MeVK = 260 MeVJ sym (crust) = 296.8 MeV FIG. 3: (Color online) The crust-core transition density as a function of K sym with the indicated three different K sym − L correlations with K = 220 ,
240 and 260 MeV and J sym (crust) = −
200 (left) and +296 . The incompressibility K of symmetric nuclear matteralso shows a significant effect especially in the case withthe Holt correlation. Its increase makes the transitiondensity higher. This can be understood easily from Eq.(8). A higher value of K needs a higher ρ t to make K µ zero as the first term of Eq. (8) involving K is alwayspositive and increases with density quadratically.It is seen that the J sym parameter plays an appreciablerole. Its increase also makes the transition density higherthrough the E − ( ρ ) term in Eq. (8). As shown in Fig.1, in the same range of L, the three correlations havedifferent ranges for the K sym parameter. Since the latterplays the most important role in determining the crust-core transition density, the correlation effects look moreobvious in Fig. 3 where the transition density is shownas a function of K sym . We notice that because the L and K sym are correlated, the results shown in Fig. 2 and Fig.3 are not independent and can be translated into each other easily.For comparisons, it is worth noting that effects of boththe K sym − L correlation and K examined here are ac-tually larger than those due to the isospin-dependence ofthe surface tension examined in the compressible liquiddrop model [42, 76] or the coefficients of the δ and δ terms in expanding the EOS of isospin-asymmetric nu-clear matter [47, 48, 67]. Thus, considering all the factorsand their associated uncertainties, it appears that themodel-dependent K sym − L correlation is a dominatingfactor in determining the crust-core transition density. B. Effects of the K sym − L correlation on thecrust-core transition pressure We notice that in constructing the EOS for the wholeNS by connecting the core EOS with that of the crust, the
40 45 50 55 60 65 70 75 80L [MeV]00.10.20.30.40.50.60.70.80.91 P t [ M e V f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017)K = 220 MeVK = 240 MeVK = 260 MeV J sym (crust) = -200 MeV 40 45 50 55 60 65 70 75 80L [MeV]00.10.20.30.40.50.60.70.80.91 P t [ M e V f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017) K = 220 MeVK = 240 MeVK = 260 MeVJ sym (crust) = 296.8 MeV FIG. 4: (Color online)The crust-core transition pressure as a function of L with the indicated three different K sym − L correlations, K = 220 ,
240 and 260 MeV, J sym (crust) = −
200 (left) and +296 . -225 -200 -175 -150 -125 -100 -75 -50 -25 0 25K sym [MeV]00.10.20.30.40.50.60.70.80.91 P t [ M e V f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017)K = 220 MeVK = 240 MeVK = 260 MeV J sym (crust) = -200 MeV -225 -200 -175 -150 -125 -100 -75 -50 -25 0 25K sym [MeV]00.10.20.30.40.50.60.70.80.91 P t [ M e V f m - ] Holt et al. (2018)Tews et al. (2017)Mondal et al. (2017)K = 220 MeVK = 240 MeVK = 260 MeVJ sym (crust) = 296.8 MeV FIG. 5: (Color online)The crust-core transition pressure as a function of K sym with the indicated three different K sym − L correlations, K = 220 ,
240 and 260 MeV, J sym (crust) = −
200 (left) and +296 . transition density plays the major role while the transi-tion pressure is only used to check if the pressure aroundthe transition point continuously increases with densityto ensure the dynamical stability of the NS. Thus, all ef-fects of the different K sym − L correlations on the radiiand tidal deformations of NSs are coming through thecrust-core transition density and the core EOS. Never-theless, it is interesting to examine how the transitionpressure itself depends on the EOS parameters especiallythe K sym − L correlation.Moreover, the crust-core transition pressure may playan important role in understanding the still puzzlingglitch phenomenon of some pulsars. The crustal frac-tion of the moment of inertia ∆ I/I is a quantity thatcan be extracted from observations of pulsar glitches. Itcan be expressed approximately in terms of the crust-core transition density and pressure as [5–7]∆ II ≈ πP t R M c (1 − . ξ − . ξ ) ξ × (cid:20) P t (1 + 5 ξ − ξ ) ρ t mc ξ (cid:21) − (13)where m is the baryon mass and ξ = GM/Rc . Analyt-ically, the K sym − L correlation may affect significantlythe ∆ I/I through both the transition density and pres-sure. It is thus also important to examine effects of the K sym − L correlation on the crust-core transition pres-sure.Having found the crust-core transition density ρ t andthe corresponding isospin asymmetry δ t through thecharge neutrality and β -equilibrium conditions [27], onecan find the corresponding crust-core transition pressure
10 11 12 13 14 15 16 17 1800.250.50.7511.251.51.7522.252.5 M / M s un L = 40 MeVL = 50 MeVL = 60 MeVL = 70 MeVL = 80 MeV
10 11 12 13 14 15 16 17 18Radius [km] 10 11 12 13 14 15 16 17 18Holt et al. (2018) Tews et al. (2017) K = 240 MeV Mondal et al. (2017)
FIG. 6: (Color online) The mass-radius correlation of neutron stars obtained using the three different K sym − L correlations asindicated but the same K = 240 MeV and all other EOS parameters indicated or given in the text. using the formalism given in Section II. Shown in Fig.4 and Fig. 5 are the crust-core transition pressures asfunctions of L and K sym , respectively. The other EOSparameters used are the same as those used in calculatingthe transition density shown in Fig. 2 and Fig. 3. It isseen that the trends follow that of the transition densityas one expects. However, effects of the K are signifi-cantly reduced while the strong effects of the K sym − L correlation and the appreciable effects of J sym remain.In the thermodynamical approach used here, the crust-core transition pressure can be approximated as [5–7] P t ≈ K ρ t ρ (cid:18) ρ t ρ − (cid:19) (14)+ ρ t δ t " − δ t E sym ( ρ t ) + (cid:18) ρ dE sym ( ρ ) dρ (cid:19) ρ t δ t . Besides the explicit dependence on the magnitude andslope of the symmetry energy at ρ t , the latter itself alsocarries effects of the symmetry energy. Effects of the var-ious EOS parameters are thus intertwined in the transi-tion pressure. Nevertheless, it is clearly seen that the K sym − L correlation plays the dominating role in deter-mining the crust-core transition pressure. C. Imprints of the K sym − L correlation on theradius of canonical neutron stars We now turn to investigating effects of the K sym − L correlation on some observational properties of canoni-cal NSs of mass 1.4 M ⊙ . We focus on the radii and tidal deformations of these canonical NSs as they are most sen-sitive to the L and K sym without much influences of thehigh-order EOS parameters, such as the J and J sym [19].The latter are known to affect significantly the massesand radii of more massive NSs [20, 27].Shown in Fig. 6 are the NS mass-radius (M-R) corre-lations obtained using the three different K sym − L cor-relations but the same K = 240 MeV and all other EOSparameters discussed earlier. Our results with other val-ues of K are similar and the resulting effects on theradius R . of canonical neutron stars will be examinedlater. Overall, the three correlations lead to generallyvery similar M-R correlations but there are interestingdifferences in the L dependence especially at low L val-ues. The increase of R . with L is a well known andcommon feature of all EOSs, see, e.g., Ref. [85].As mentioned earlier, the maximum mass of neutronstars is mostly determined by the SNM EOS character-ized by the K and J parameters. While effects of thesymmetry energy parameters on the maximum mass aregenerally small but can be appreciable when the K and J are fixed. Comparing the results in the three windowswith different K sym − L correlations, it is seen that boththe maximum mass and the radius R . are appreciablydifferent at low L values. This can be well understoodfrom the K sym − L correlations shown in Fig. 1. For lowL values, the K sym values are the lowest for the Mondalcorrelation but the highest for the Holt case. The latterthus has the stiffest while the Modal case has the softestsymmetry energy, making the strongest and the weakestcontribution to the nuclear pressure, respectively. Con-sequently, for low L values, the Holt correlation predictsa higher value for the maximum mass and also a larger
40 45 50 55 60 65 70 75 801010.210.410.610.81111.211.411.611.81212.212.412.612.81313.213.413.613.814 R . [ k m ] K = 220 MeVK = 240 MeVK = 260 MeV40 45 50 55 60 65 70 75 80L [MeV] 40 45 50 55 60 65 70 75 80Holt et al. (2018) Tews et al. (2017) Mondal et al. (2017) FIG. 7: (Color online) The radius R . of canonical neutron stars of mass 1.4 M ⊙ as a function of the symmetry energy slopeparameter L obtained using the three different K sym − L correlations as indicated and three different values of K while keepingall other EOS parameters the same as indicated or given in the text. The magenta box indicate the most probable value of R . = 11 . +0 . − . km at 90% confidence level from the latest multimessenger observation of GW170817 [86]. radius compared to the other two cases. While it is theopposite for the Mondal case. As we discussed in SectionII C, in the Mondal and Tew cases, the second dynam-ical instability may happen at high densities when theresulting symmetry energy is super-soft with small L val-ues but big negative K sym values especially if K is alsosmall. Without introducing new phases to stabilize neu-tron star matter above the onset density of the seconddynamical instability, the maximum mass supported bythe Mondal and Tews correlations are thus smaller com-pared to the Holt result in cases where both the L and K values are small towards their lower limits currentlyknown.The effects of different K sym − L correlations on theradius R . of canonical NSs can be seen more clearlyin Fig. 7 where the R . is shown as a function of Lwith three different K values covering their current un-certainty ranges. As mentioned earlier, the high-densityEOS parameters J and J sym have little effects on the R . [19, 20, 27–29]. We thus focus on the dependenceof R . on L and K using the three different K sym − L correlations. A recent study combining multimessengerobservations of GW170817 and many-body theory pre-dictions using nuclear forces based on the chiral effec-tive field theory found that the most probable R . is R . = 11 . +0 . − . km at 90% confidence level [86]. Thelatter is indicated with the magenta boxes in Fig. 7 forcomparisons.Several interesting observations can be made fromcomparing the results in the three windows: • The R . increases almost linearly with both L and K in their respective uncertainty ranges. The dualdependence of R . on L and K indicates that it isimportant to have the prior knowledge of K as ac-curately as possible to infer precisely the value of Lfrom NS radius measurements. On the other hand,it has been known for a long time that the remain-ing uncertainty of about ±
20 MeV in extracting K from studying giant resonances of finite nucleiis mainly due to the correlations of K and param-eters characterizing the E sym ( ρ ) near ρ , see, e.g.,Refs. [87, 88] for detailed discussions. It is thus in-teresting to note that Bayesian inferences of multi-ple EOS parameters simultaneously from combineddata of astrophysical observations and nuclear ex-periments have the potential to further pin downthe L and K parameters all together [19, 20]. • For all three K sym − L correlations considered, the R . is about the same when L is higher than about60 MeV. This is because at higher L values, thethree K sym − L correlations largely overlap as shownin Fig. 1. The R . becomes gradually more dif-ferent as the L decreases. The Holt correlation hasthe stiffest symmetry energy leading to the largestradius while the Mondal case has the softest sym-metry energy giving the smallest R . value as wediscussed earlier. Moreover, effects of K on R . is almost independent of the K sym − L correlation.At a given L value, the R . increases with increas-ing K as one expects. Quantitatively, however,0the uncertainty of R . due to that of K is onlyabout 4% while that due to the uncertainty of L isabout 16%. For a comparison, at L=40 MeV with K = 220 MeV, the difference in R . from usingthe Holt and Mondal K sym − L correlation is about12%. This is actually slightly less than the ap-proximately 14% uncertainty of the latest constrainon R . from the multimessengers observations ofGW170817. Thus, the latter can not distinguishthe three K sym − L correlations. • The observational constraint R . = 11 . +0 . − . kmcan put some useful limits on the slope L. The Holtcorrelation requires the L parameter to be less thanabout 45 MeV close to the lower boundary of itsfiducial value. While the other two K sym − L cor-relations prefers an upper limit of L in the range of45-55 MeV for K between 260 and 220 MeV. Thisinformation is useful for more accurate extractionof EOS parameters and understanding their modeldependences in future analyses of more data to beavailable.
40 45 50 55 60 65 70 75 80L [MeV]1010.210.410.610.81111.211.411.611.81212.212.412.612.81313.213.413.613.814 R . [ k m ] J sym (crust) = J sym (core) = 296.8 MeVJ sym (crust) = -200 MeV and J sym (core) = 296.8 MeV Mondal et al. (2017)K = 220 MeVK = 240 MeVK = 260 MeV FIG. 8: (Color online) Solid: The default calculation (withthe same J sym of 296.8 MeV for the core EOS and for findingthe crust-core transition point). Dashed: a calculation using J sym = −
200 MeV for finding the crust-core transition pointbut still the default J sym for the core EOS with the Mondalcorrelation. The K sym − L correlation affects both the crust-coretransition point and the core EOS. We have shown ear-lier how the crust-core transition density and pressure arealso being affected appreciably by the uncertain J sym pa-rameter characterizing the E sym ( ρ ) far away from ρ ateither sub-saturation or supra-saturation densities [20].More quantatively, from J sym = −
200 MeV to +296 . J sym on the R . we have donesystematical studies by using different combinations of J sym parameters in calculating the crust-core transitionpoint and the core EOS. We found that the effects arenegligibly small except when the L value is small close tothe lower boundary of its fiducial value. As an example,shown in Fig. 8 is a comparison of the default calcu-lation (with the same J sym of 296.8 MeV for the coreEOS and for finding the crust-core transition point) anda calculation that uses J sym = −
200 MeV for finding thecrust-core transition point (while the core EOS still usesthe default J sym of 296.8 MeV) with the Mondal corre-lation. It is seen that the R . versus L results from thetwo calculation are not much different except when theL becomes smaller than about 50 MeV and K is alsosmall. When the L becomes small, its contribution tothe pressure is smaller. Then, what values one use forthe high-order symmetry energy parameter J sym in find-ing where to connect the core EOS to the crust EOS be-comes more important. While this is the same region of Lwhere the K sym − L correlation plays a significant role indetermining the R . , obviously the K sym − L correlationeffect is much stronger than that due to uncertainty ofthe J sym parameter. Moreover, the comparison indicatesthat the K sym − L correlation effects on the R . in thedefault calculations shown in Figs. 6 and 7 come mainlythrough the core EOS. D. Imprints of the K sym − L correlation on the tidaldeformation and its correlation with the radius ofcanonical neutron stars We now examine imprints of the K sym − L correlationon neutron stars’ scaled tidal deformabilityΛ = 23 k [( c /G ) R/M ] (15)where k is the second Love number obtained from solv-ing coupled differential equations simultaneously withthe TOV equation [89–92]. Shown in Fig. 9 is Λ as a func-tion of mass obtained using the three different K sym − L correlations but all the same EOS parameters. The ma-genta bar between 70 and 580 is the tidal deformabilityof canonical neutron stars from LIGO/VIRGO’s obser-vation of GW170817 [16]. Similar to the mass-radiuscorrelation shown in Fig. 6, the K sym − L correlationhas some observable influences on the tidal deformabil-ity for small L values. Again, it can be easily explainedby the different K sym values with the different K sym − L correlation when L is small, as shown in Fig. 1. Unfortu-nately, the range of Λ extracted by LIGO/VIRGO fromGW170817 is still too big to set a firmer limit on L thanits fiducial range. It also can not differentiate the differ-ent K sym − L correlations. Namely, the use of different K sym − L correlations does not affect what one extractsabout the symmetry energy from the observational con-straint on Λ alone from GW170817.Effects of the K sym − L correlation can be more clearlyseen in the R − Λ correlation by combing the M − R plot1 Λ L = 40 MeVL = 50 MeVL = 60 MeVL = 70 MeVL = 80 MeV sun K = 240 MeV Mondal et al.GW170817
FIG. 9: (Color online) The scaled tidal deformability as a function of mass of neutron stars obtained using the three different K sym − L correlations as indicated but the same K = 240 MeV and all other EOS parameters indicated or given in the text.The magenta bar between 70 and 580 is the tidal deformability of canonical neutron stars extracted from LIGO/VIRGO’sobservation of GW170817 [16]. R a d i u s [ k m ] L = 40 MeVL = 50 MeVL = 60 MeVL = 70 MeVL = 80 MeV 0 200 400 600 800 Λ sun = 240 MeV Mondal et al. (2017) FIG. 10: (Color online) The radius-tidal deformability correlations of neutron stars with all masses obtained using the threedifferent K sym − L correlations as indicated but the same K = 240 MeV. The magenta circles linked on the dashed lines arefor canonical neutron stars. of Fig. 6 and the Λ − M plot of Fig. 9. In the R − Λ corre-lation plot shown in Fig. 10, each point on a given curewith a fixed L has a specific mass. Starting from thefirst point on the left, the mass decreases continuouslyon any curve with the same L. For the very massive neu- tron stars with masses around 2.0 to 1.8 M ⊙ on eachcurve the radius increases monotonically with increasingΛ (decreasing mass) until the plateau is reached. On theplateau, namely in a large range of mass around 1.4 M ⊙ from approximately 1.8 down to 0.5 M ⊙ , the radius stays2approximately a constant while the Λ keeps increasing.The magenta circles linked on the dashed lines are forcanonical neutron stars of mass 1.4 M ⊙ on curves withdifferent L values. It is seen that for these canonical NSsof the same mass, the radius increases with Λ approxi-mately linearly. It is known that the upper limit of Λ forcanonical NSs from observing GW170817 is more reliablethan its lower limit which is more model dependent [16].While the two NSs involved in GW170817 have signifi-cantly different mass ranges, two independent analyses[16, 93] using different approaches all found consistentlythat the two NSs have essentially identical radii. It isthus appropriate for the discussions here to assume theNS mass is 1.4 M ⊙ . Comparing results in the three win-dows and using the Λ maximum = 580, it is seen that allthree K sym − L correlations give consistently the same up-per limit of R . ≤ . L ≤
80 MeV. This is consistent with the resultsof recent Bayesian analyses [19, 20] of the LIGO/VIRGOdata as we mentioned in the introduction. Again, as theL decreases, effects of the K sym − L correlation becomesmore obvious. Unfortunately, the lower limit of Λ fromGW170817 is currently unreliable. In fact, the lower limitof Λ minimum = 70 requires L values much smaller thanthe lower boundary of its fiducial value discussed in theintroduction.In principle, independent measurements of R . andΛ . for canonical NSs will put a more stringent con-straint on L and possibly also on the K sym − L correlation.While many recent works in the literature have focusedon extracting the R . from Λ . using the GW170817data, there are independent measurements of R . fromother observations, such as X-ray observations. Since theconstraining box on R . shown in Fig. 10 already usedthe Λ from GW170817 as one of the multimessengers, letsexamine how the NICER’s recent measurement of PSRJ0030+0451 using X-rays may help. The NICER Col-laboration measured simultaneously both the mass andradius of PSR J0030+0451. Their results from two some-what independent analyses are: M = 1 . +0 . − . M ⊙ and R = 13 . +1 . − . km [18], and M = 1 . +0 . − . M ⊙ and R = 12 . +1 . − . km [17] at 68% confidence level. For aqualitative discussion, one can safely assume the mass isabout 1.4 M ⊙ . The most probable radii from both analy-ses are consistent with the upper radius indicated by theupper limit of Λ . from LIGO/VIRGO as we discussedabove. However, NICER’s 68% upper radius bound-ary is as high as 14.26 km or 13.9 km, allowing muchhigher L values than that allowed by the LIGO/VIRGOdata beyond the limit of its fiducial value. Nevertheless,NICER’s lower radius limit R . (minimum) from the twoanalyses, i.e., R . (minimum)=11.96 km or 11.57 km, canput a useful lower limit L minimum on L. This limit, how-ever, depends on the K sym − L correlation one uses. It isseen from Fig. 10 that L minimum is between 40-50 MeV,50-60 MeV and 50-60 MeV for the Holt, Tews and Mon-dal correlations, respectively. As we discussed before, thelast two have approximately the same K sym − L corre- lation from the same sets of model predictions. Thus,the different K sym − L correlations considered affect theextraction of L minimum by above 10 MeV.Overall, the most probable value of R . from NICERcan independently limit the most probable value of L tothe range of 40-80 MeV consistent with its known fiducialrange. This is also consistent with the finding of the de-tailed Bayesian analyses of the combined LIGO/VIRGOand NICER data [20]. Certainly, more coming indepen-dent data for both the R . and Λ . will help furtherconstrain the value of L and the K sym − L correlation. IV. SUMMARY AND CONCLUSIONS
In summary, using a meta-model of nuclear EOSs weexamined effects of nuclear EOS parameters especiallythe curvature ( K sym )-slope (L) correlation of nuclearsymmetry energy on the crust-core transition densityand pressure in neutron stars. We also examinedimprints of the K sym − L correlation on astrophysicalobservables especially the radius and tidal deformabilityof canonical neutron stars.Our main conclusions are the following: • The crust-core transition density and pressure havesome appreciable dependences on the incompress-ibility K but are insensitive to the skewness J ofsymmetric nuclear matter. • The crust-core transition density and pressure aresensitive to the L , K sym and J sym parameters in-dependently as well as the K sym − L correlation. • The curvature K sym plays a more important rolethan the slope L in determining the crust-core tran-sition density. • The J and J sym parameters have little effects on R . and the K sym − L correlation effects comethrough the core EOS. • The K sym − L correlation has strong imprints on theradius and tidal deformability of canonical neutronstars especially when the slope L is close to thelower limit (40 MeV) of its currently known fiducialvalue.The astrophysical imprints of K sym − L correlationcan potentially help better constrain the poorly knownhigh-density behavior of nuclear symmetry energy. Inparticular, if a unique K sym − L correlation can be firmlyestablished by observations/experiments, it will facilitatethe extraction of the very poorly known K sym parameterprogressively from the relatively better determined Lvalue. We thus also examined whether existing tidaldeformability data from LIGO/VIRGO’s observation ofGW170817 an the NS radius data from NICER’s recentobservation of PSR J0030+0451 can help distinguish the3three different K sym − L correlations considered, and howwell they can constrain the L parameter. Consistentlywith earlier findings in the literature, they can put someuseful constraints on L. Unfortunately, they can notdistinguish the three different K sym − L correlationsstudied. Nevertheless, more precise measurements ofespecially independent radius and tidal deformabilitydata from multiple observables holds the strong promiseof pinning down the curvature-slope correlation, thushelp constrain the high-density behavior of nuclearsymmetry energy. Acknowledgments
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